Well,...yes, you have the right idea,...What hasn't been discussed,and for the most part doesn't really need to be given what we are dealing with,....is roll off. I'm not sure what the roll off of the crossover in the anthem is,...but for practical explanation,...The crossover doesn't cut off output at a specific frequency,....it starts at that level, and slowly lessens the frequencies above or below that point, depending on weather its a high-pass or low-pass crossover.
There again, yes...If you cross over your mains at say 100,..and the sub at 40, then yes, you could realize a hole, so to speak in between the two crossover points.
Again, given what you have there, all small speakers accept the sub, you could cross them all over at 80,....High-pass for the mains and surrounds, and low-pass for the sub. If you can't turn the sub crossover off, then turn it all the way up, and let the Anthem do the work.
The AVM20 is very tweekable, and will allow you to set about any system up anyway you like,...Each speaker individually if you like,....Not sure exactly what speakers you have, but you have many options with the Anthem,...Dig deep enough in that monster of a manual, and you will find, that machine will all but give you a massage while your listening to your favorite music,....;)